Quadratic roots, medium range

Percentage Accurate: 31.4% → 99.4%

Time: 9.9s

Alternatives: 8

Speedup: 3.6×

Specification

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 31.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}{\left(\sqrt{\mathsf{fma}\left(a \cdot \frac{c}{b \cdot b}, -4, 1\right) \cdot \left(b \cdot b\right)} + b\right) \cdot \left(2 \cdot a\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (fma (* -4.0 a) c 0.0)
  (* (+ (sqrt (* (fma (* a (/ c (* b b))) -4.0 1.0) (* b b))) b) (* 2.0 a))))

C

double code(double a, double b, double c) {
	return fma((-4.0 * a), c, 0.0) / ((sqrt((fma((a * (c / (b * b))), -4.0, 1.0) * (b * b))) + b) * (2.0 * a));
}

Julia

function code(a, b, c)
	return Float64(fma(Float64(-4.0 * a), c, 0.0) / Float64(Float64(sqrt(Float64(fma(Float64(a * Float64(c / Float64(b * b))), -4.0, 1.0) * Float64(b * b))) + b) * Float64(2.0 * a)))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(-4.0 * a), $MachinePrecision] * c + 0.0), $MachinePrecision] / N[(N[(N[Sqrt[N[(N[(N[(a * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -4.0 + 1.0), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision] * N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.6%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\left(4 \cdot a\right) \cdot c}}}{2 \cdot a} \]

   3. fp-cancel-sub-sign-inv N/A

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{b \cdot b + \left(\mathsf{neg}\left(4 \cdot a\right)\right) \cdot c}}}{2 \cdot a} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{b \cdot b} + \left(\mathsf{neg}\left(4 \cdot a\right)\right) \cdot c}}{2 \cdot a} \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, \left(\mathsf{neg}\left(4 \cdot a\right)\right) \cdot c\right)}}}{2 \cdot a} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(\mathsf{neg}\left(4 \cdot a\right)\right) \cdot c}\right)}}{2 \cdot a} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(\mathsf{neg}\left(\color{blue}{4 \cdot a}\right)\right) \cdot c\right)}}{2 \cdot a} \]

   8. distribute-lft-neg-in N/A

      \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(\left(\mathsf{neg}\left(4\right)\right) \cdot a\right)} \cdot c\right)}}{2 \cdot a} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(\left(\mathsf{neg}\left(4\right)\right) \cdot a\right)} \cdot c\right)}}{2 \cdot a} \]

   10. metadata-eval 31.7

       \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(\color{blue}{-4} \cdot a\right) \cdot c\right)}}{2 \cdot a} \]

4. Applied rewrites 31.7%

   \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)}}}{2 \cdot a} \]
5. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)}}}{2 \cdot a} \]

   2. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} + \left(-b\right)}}{2 \cdot a} \]

   3. flip-+ N/A

      \[\leadsto \frac{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} \cdot \sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} - \left(-b\right) \cdot \left(-b\right)}{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} - \left(-b\right)}}}{2 \cdot a} \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} \cdot \sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} - \left(-b\right) \cdot \left(-b\right)}{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} - \left(-b\right)}}}{2 \cdot a} \]

6. Applied rewrites 32.5%

   \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) - b \cdot b}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}}{2 \cdot a} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) - b \cdot b}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}{2 \cdot a}} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) - b \cdot b}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}}{2 \cdot a} \]

   3. associate-/l/ N/A

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) - b \cdot b}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)}} \]

   4. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) - b \cdot b}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)}} \]

   5. lift--.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) - b \cdot b}}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)} \]

   6. lift-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(\left(-4 \cdot a\right) \cdot c + b \cdot b\right)} - b \cdot b}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)} \]

   7. associate--l+ N/A

      \[\leadsto \frac{\color{blue}{\left(-4 \cdot a\right) \cdot c + \left(b \cdot b - b \cdot b\right)}}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b - b \cdot b\right)}}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)} \]

   9. +-inverses N/A

      \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot a, c, \color{blue}{0}\right)}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)} \]

   10. lower-*.f64 99.4

       \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}{\color{blue}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)}} \]

8. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)}} \]

9. Taylor expanded in b around inf

   \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}{\left(\sqrt{\color{blue}{{b}^{2} \cdot \left(1 + -4 \cdot \frac{a \cdot c}{{b}^{2}}\right)}} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)} \]
10. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}{\left(\sqrt{\color{blue}{\left(1 + -4 \cdot \frac{a \cdot c}{{b}^{2}}\right) \cdot {b}^{2}}} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)} \]

    2. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}{\left(\sqrt{\color{blue}{\left(1 + -4 \cdot \frac{a \cdot c}{{b}^{2}}\right) \cdot {b}^{2}}} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)} \]

    3. +-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}{\left(\sqrt{\color{blue}{\left(-4 \cdot \frac{a \cdot c}{{b}^{2}} + 1\right)} \cdot {b}^{2}} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)} \]

    4. *-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}{\left(\sqrt{\left(\color{blue}{\frac{a \cdot c}{{b}^{2}} \cdot -4} + 1\right) \cdot {b}^{2}} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)} \]

    5. lower-fma.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}{\left(\sqrt{\color{blue}{\mathsf{fma}\left(\frac{a \cdot c}{{b}^{2}}, -4, 1\right)} \cdot {b}^{2}} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)} \]

    6. associate-/l* N/A

       \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}{\left(\sqrt{\mathsf{fma}\left(\color{blue}{a \cdot \frac{c}{{b}^{2}}}, -4, 1\right) \cdot {b}^{2}} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)} \]

    7. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}{\left(\sqrt{\mathsf{fma}\left(\color{blue}{a \cdot \frac{c}{{b}^{2}}}, -4, 1\right) \cdot {b}^{2}} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)} \]

    8. lower-/.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}{\left(\sqrt{\mathsf{fma}\left(a \cdot \color{blue}{\frac{c}{{b}^{2}}}, -4, 1\right) \cdot {b}^{2}} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)} \]

    9. unpow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}{\left(\sqrt{\mathsf{fma}\left(a \cdot \frac{c}{\color{blue}{b \cdot b}}, -4, 1\right) \cdot {b}^{2}} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)} \]

    10. lower-*.f64 N/A

        \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}{\left(\sqrt{\mathsf{fma}\left(a \cdot \frac{c}{\color{blue}{b \cdot b}}, -4, 1\right) \cdot {b}^{2}} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)} \]

    11. unpow2 N/A

        \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}{\left(\sqrt{\mathsf{fma}\left(a \cdot \frac{c}{b \cdot b}, -4, 1\right) \cdot \color{blue}{\left(b \cdot b\right)}} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)} \]

    12. lower-*.f64 99.4

        \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}{\left(\sqrt{\mathsf{fma}\left(a \cdot \frac{c}{b \cdot b}, -4, 1\right) \cdot \color{blue}{\left(b \cdot b\right)}} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)} \]

11. Applied rewrites 99.4%

    \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}{\left(\sqrt{\color{blue}{\mathsf{fma}\left(a \cdot \frac{c}{b \cdot b}, -4, 1\right) \cdot \left(b \cdot b\right)}} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)} \]

12. Final simplification 99.4%

    \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}{\left(\sqrt{\mathsf{fma}\left(a \cdot \frac{c}{b \cdot b}, -4, 1\right) \cdot \left(b \cdot b\right)} + b\right) \cdot \left(2 \cdot a\right)} \]
13. Add Preprocessing

Alternative 2: 99.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b\right) \cdot \left(a + a\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (fma (* -4.0 a) c 0.0)
  (* (+ (sqrt (fma (* -4.0 a) c (* b b))) b) (+ a a))))

C

double code(double a, double b, double c) {
	return fma((-4.0 * a), c, 0.0) / ((sqrt(fma((-4.0 * a), c, (b * b))) + b) * (a + a));
}

Julia

function code(a, b, c)
	return Float64(fma(Float64(-4.0 * a), c, 0.0) / Float64(Float64(sqrt(fma(Float64(-4.0 * a), c, Float64(b * b))) + b) * Float64(a + a)))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(-4.0 * a), $MachinePrecision] * c + 0.0), $MachinePrecision] / N[(N[(N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision] * N[(a + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.6%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\left(4 \cdot a\right) \cdot c}}}{2 \cdot a} \]

   3. fp-cancel-sub-sign-inv N/A

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{b \cdot b + \left(\mathsf{neg}\left(4 \cdot a\right)\right) \cdot c}}}{2 \cdot a} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{b \cdot b} + \left(\mathsf{neg}\left(4 \cdot a\right)\right) \cdot c}}{2 \cdot a} \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, \left(\mathsf{neg}\left(4 \cdot a\right)\right) \cdot c\right)}}}{2 \cdot a} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(\mathsf{neg}\left(4 \cdot a\right)\right) \cdot c}\right)}}{2 \cdot a} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(\mathsf{neg}\left(\color{blue}{4 \cdot a}\right)\right) \cdot c\right)}}{2 \cdot a} \]

   8. distribute-lft-neg-in N/A

      \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(\left(\mathsf{neg}\left(4\right)\right) \cdot a\right)} \cdot c\right)}}{2 \cdot a} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(\left(\mathsf{neg}\left(4\right)\right) \cdot a\right)} \cdot c\right)}}{2 \cdot a} \]

   10. metadata-eval 31.7

       \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(\color{blue}{-4} \cdot a\right) \cdot c\right)}}{2 \cdot a} \]

4. Applied rewrites 31.7%

   \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)}}}{2 \cdot a} \]
5. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)}}}{2 \cdot a} \]

   2. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} + \left(-b\right)}}{2 \cdot a} \]

   3. flip-+ N/A

      \[\leadsto \frac{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} \cdot \sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} - \left(-b\right) \cdot \left(-b\right)}{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} - \left(-b\right)}}}{2 \cdot a} \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} \cdot \sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} - \left(-b\right) \cdot \left(-b\right)}{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} - \left(-b\right)}}}{2 \cdot a} \]

6. Applied rewrites 32.5%

   \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) - b \cdot b}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}}{2 \cdot a} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) - b \cdot b}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}{2 \cdot a}} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) - b \cdot b}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}}{2 \cdot a} \]

   3. associate-/l/ N/A

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) - b \cdot b}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)}} \]

   4. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) - b \cdot b}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)}} \]

   5. lift--.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) - b \cdot b}}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)} \]

   6. lift-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(\left(-4 \cdot a\right) \cdot c + b \cdot b\right)} - b \cdot b}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)} \]

   7. associate--l+ N/A

      \[\leadsto \frac{\color{blue}{\left(-4 \cdot a\right) \cdot c + \left(b \cdot b - b \cdot b\right)}}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b - b \cdot b\right)}}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)} \]

   9. +-inverses N/A

      \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot a, c, \color{blue}{0}\right)}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)} \]

   10. lower-*.f64 99.4

       \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}{\color{blue}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)}} \]

8. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)}} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)\right) \cdot \color{blue}{\left(2 \cdot a\right)}} \]

   2. count-2-rev N/A

      \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)\right) \cdot \color{blue}{\left(a + a\right)}} \]

   3. lower-+.f64 99.4

      \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)\right) \cdot \color{blue}{\left(a + a\right)}} \]

10. Applied rewrites 99.4%

    \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)\right) \cdot \color{blue}{\left(a + a\right)}} \]

11. Final simplification 99.4%

    \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b\right) \cdot \left(a + a\right)} \]
12. Add Preprocessing

Alternative 3: 90.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot c}{b}, -4, \left(b \cdot a\right) \cdot 4\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (fma (* -4.0 a) c 0.0) (fma (/ (* (* a a) c) b) -4.0 (* (* b a) 4.0))))

C

double code(double a, double b, double c) {
	return fma((-4.0 * a), c, 0.0) / fma((((a * a) * c) / b), -4.0, ((b * a) * 4.0));
}

Julia

function code(a, b, c)
	return Float64(fma(Float64(-4.0 * a), c, 0.0) / fma(Float64(Float64(Float64(a * a) * c) / b), -4.0, Float64(Float64(b * a) * 4.0)))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(-4.0 * a), $MachinePrecision] * c + 0.0), $MachinePrecision] / N[(N[(N[(N[(a * a), $MachinePrecision] * c), $MachinePrecision] / b), $MachinePrecision] * -4.0 + N[(N[(b * a), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.6%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\left(4 \cdot a\right) \cdot c}}}{2 \cdot a} \]

   3. fp-cancel-sub-sign-inv N/A

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{b \cdot b + \left(\mathsf{neg}\left(4 \cdot a\right)\right) \cdot c}}}{2 \cdot a} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{b \cdot b} + \left(\mathsf{neg}\left(4 \cdot a\right)\right) \cdot c}}{2 \cdot a} \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, \left(\mathsf{neg}\left(4 \cdot a\right)\right) \cdot c\right)}}}{2 \cdot a} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(\mathsf{neg}\left(4 \cdot a\right)\right) \cdot c}\right)}}{2 \cdot a} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(\mathsf{neg}\left(\color{blue}{4 \cdot a}\right)\right) \cdot c\right)}}{2 \cdot a} \]

   8. distribute-lft-neg-in N/A

      \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(\left(\mathsf{neg}\left(4\right)\right) \cdot a\right)} \cdot c\right)}}{2 \cdot a} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(\left(\mathsf{neg}\left(4\right)\right) \cdot a\right)} \cdot c\right)}}{2 \cdot a} \]

   10. metadata-eval 31.7

       \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(\color{blue}{-4} \cdot a\right) \cdot c\right)}}{2 \cdot a} \]

4. Applied rewrites 31.7%

   \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)}}}{2 \cdot a} \]
5. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)}}}{2 \cdot a} \]

   2. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} + \left(-b\right)}}{2 \cdot a} \]

   3. flip-+ N/A

      \[\leadsto \frac{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} \cdot \sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} - \left(-b\right) \cdot \left(-b\right)}{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} - \left(-b\right)}}}{2 \cdot a} \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} \cdot \sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} - \left(-b\right) \cdot \left(-b\right)}{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} - \left(-b\right)}}}{2 \cdot a} \]

6. Applied rewrites 32.5%

   \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) - b \cdot b}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}}{2 \cdot a} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) - b \cdot b}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}{2 \cdot a}} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) - b \cdot b}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}}{2 \cdot a} \]

   3. associate-/l/ N/A

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) - b \cdot b}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)}} \]

   4. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) - b \cdot b}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)}} \]

   5. lift--.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) - b \cdot b}}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)} \]

   6. lift-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(\left(-4 \cdot a\right) \cdot c + b \cdot b\right)} - b \cdot b}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)} \]

   7. associate--l+ N/A

      \[\leadsto \frac{\color{blue}{\left(-4 \cdot a\right) \cdot c + \left(b \cdot b - b \cdot b\right)}}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b - b \cdot b\right)}}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)} \]

   9. +-inverses N/A

      \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot a, c, \color{blue}{0}\right)}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)} \]

   10. lower-*.f64 99.4

       \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}{\color{blue}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)}} \]

8. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)}} \]

9. Taylor expanded in c around 0

   \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}{\color{blue}{-4 \cdot \frac{{a}^{2} \cdot c}{b} + 4 \cdot \left(a \cdot b\right)}} \]
10. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}{\color{blue}{\frac{{a}^{2} \cdot c}{b} \cdot -4} + 4 \cdot \left(a \cdot b\right)} \]

    2. lower-fma.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}{\color{blue}{\mathsf{fma}\left(\frac{{a}^{2} \cdot c}{b}, -4, 4 \cdot \left(a \cdot b\right)\right)}} \]

    3. lower-/.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}{\mathsf{fma}\left(\color{blue}{\frac{{a}^{2} \cdot c}{b}}, -4, 4 \cdot \left(a \cdot b\right)\right)} \]

    4. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}{\mathsf{fma}\left(\frac{\color{blue}{{a}^{2} \cdot c}}{b}, -4, 4 \cdot \left(a \cdot b\right)\right)} \]

    5. unpow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}{\mathsf{fma}\left(\frac{\color{blue}{\left(a \cdot a\right)} \cdot c}{b}, -4, 4 \cdot \left(a \cdot b\right)\right)} \]

    6. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}{\mathsf{fma}\left(\frac{\color{blue}{\left(a \cdot a\right)} \cdot c}{b}, -4, 4 \cdot \left(a \cdot b\right)\right)} \]

    7. *-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot c}{b}, -4, \color{blue}{\left(a \cdot b\right) \cdot 4}\right)} \]

    8. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot c}{b}, -4, \color{blue}{\left(a \cdot b\right) \cdot 4}\right)} \]

    9. *-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot c}{b}, -4, \color{blue}{\left(b \cdot a\right)} \cdot 4\right)} \]

    10. lower-*.f64 90.9

        \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot c}{b}, -4, \color{blue}{\left(b \cdot a\right)} \cdot 4\right)} \]

11. Applied rewrites 90.9%

    \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}{\color{blue}{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot c}{b}, -4, \left(b \cdot a\right) \cdot 4\right)}} \]
12. Add Preprocessing

Alternative 4: 90.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}{\left(\mathsf{fma}\left(a \cdot \frac{c}{b}, -2, b\right) + b\right) \cdot \left(2 \cdot a\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (fma (* -4.0 a) c 0.0) (* (+ (fma (* a (/ c b)) -2.0 b) b) (* 2.0 a))))

C

double code(double a, double b, double c) {
	return fma((-4.0 * a), c, 0.0) / ((fma((a * (c / b)), -2.0, b) + b) * (2.0 * a));
}

Julia

function code(a, b, c)
	return Float64(fma(Float64(-4.0 * a), c, 0.0) / Float64(Float64(fma(Float64(a * Float64(c / b)), -2.0, b) + b) * Float64(2.0 * a)))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(-4.0 * a), $MachinePrecision] * c + 0.0), $MachinePrecision] / N[(N[(N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] * -2.0 + b), $MachinePrecision] + b), $MachinePrecision] * N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.6%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\left(4 \cdot a\right) \cdot c}}}{2 \cdot a} \]

   3. fp-cancel-sub-sign-inv N/A

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{b \cdot b + \left(\mathsf{neg}\left(4 \cdot a\right)\right) \cdot c}}}{2 \cdot a} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{b \cdot b} + \left(\mathsf{neg}\left(4 \cdot a\right)\right) \cdot c}}{2 \cdot a} \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, \left(\mathsf{neg}\left(4 \cdot a\right)\right) \cdot c\right)}}}{2 \cdot a} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(\mathsf{neg}\left(4 \cdot a\right)\right) \cdot c}\right)}}{2 \cdot a} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(\mathsf{neg}\left(\color{blue}{4 \cdot a}\right)\right) \cdot c\right)}}{2 \cdot a} \]

   8. distribute-lft-neg-in N/A

      \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(\left(\mathsf{neg}\left(4\right)\right) \cdot a\right)} \cdot c\right)}}{2 \cdot a} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(\left(\mathsf{neg}\left(4\right)\right) \cdot a\right)} \cdot c\right)}}{2 \cdot a} \]

   10. metadata-eval 31.7

       \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(\color{blue}{-4} \cdot a\right) \cdot c\right)}}{2 \cdot a} \]

4. Applied rewrites 31.7%

   \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)}}}{2 \cdot a} \]
5. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)}}}{2 \cdot a} \]

   2. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} + \left(-b\right)}}{2 \cdot a} \]

   3. flip-+ N/A

      \[\leadsto \frac{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} \cdot \sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} - \left(-b\right) \cdot \left(-b\right)}{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} - \left(-b\right)}}}{2 \cdot a} \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} \cdot \sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} - \left(-b\right) \cdot \left(-b\right)}{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} - \left(-b\right)}}}{2 \cdot a} \]

6. Applied rewrites 32.5%

   \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) - b \cdot b}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}}{2 \cdot a} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) - b \cdot b}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}{2 \cdot a}} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) - b \cdot b}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}}{2 \cdot a} \]

   3. associate-/l/ N/A

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) - b \cdot b}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)}} \]

   4. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) - b \cdot b}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)}} \]

   5. lift--.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) - b \cdot b}}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)} \]

   6. lift-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(\left(-4 \cdot a\right) \cdot c + b \cdot b\right)} - b \cdot b}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)} \]

   7. associate--l+ N/A

      \[\leadsto \frac{\color{blue}{\left(-4 \cdot a\right) \cdot c + \left(b \cdot b - b \cdot b\right)}}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b - b \cdot b\right)}}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)} \]

   9. +-inverses N/A

      \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot a, c, \color{blue}{0}\right)}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)} \]

   10. lower-*.f64 99.4

       \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}{\color{blue}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)}} \]

8. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)}} \]

9. Taylor expanded in a around 0

   \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}{\left(\color{blue}{\left(b + -2 \cdot \frac{a \cdot c}{b}\right)} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)} \]
10. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}{\left(\color{blue}{\left(-2 \cdot \frac{a \cdot c}{b} + b\right)} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)} \]

    2. *-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}{\left(\left(\color{blue}{\frac{a \cdot c}{b} \cdot -2} + b\right) - \left(-b\right)\right) \cdot \left(2 \cdot a\right)} \]

    3. lower-fma.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}{\left(\color{blue}{\mathsf{fma}\left(\frac{a \cdot c}{b}, -2, b\right)} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)} \]

    4. associate-/l* N/A

       \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}{\left(\mathsf{fma}\left(\color{blue}{a \cdot \frac{c}{b}}, -2, b\right) - \left(-b\right)\right) \cdot \left(2 \cdot a\right)} \]

    5. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}{\left(\mathsf{fma}\left(\color{blue}{a \cdot \frac{c}{b}}, -2, b\right) - \left(-b\right)\right) \cdot \left(2 \cdot a\right)} \]

    6. lower-/.f64 91.0

       \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}{\left(\mathsf{fma}\left(a \cdot \color{blue}{\frac{c}{b}}, -2, b\right) - \left(-b\right)\right) \cdot \left(2 \cdot a\right)} \]

11. Applied rewrites 91.0%

    \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}{\left(\color{blue}{\mathsf{fma}\left(a \cdot \frac{c}{b}, -2, b\right)} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)} \]

12. Final simplification 91.0%

    \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}{\left(\mathsf{fma}\left(a \cdot \frac{c}{b}, -2, b\right) + b\right) \cdot \left(2 \cdot a\right)} \]
13. Add Preprocessing

Alternative 5: 90.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}{\mathsf{fma}\left(a \cdot \frac{c}{b}, -4, 4 \cdot b\right) \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (fma (* -4.0 a) c 0.0) (* (fma (* a (/ c b)) -4.0 (* 4.0 b)) a)))

C

double code(double a, double b, double c) {
	return fma((-4.0 * a), c, 0.0) / (fma((a * (c / b)), -4.0, (4.0 * b)) * a);
}

Julia

function code(a, b, c)
	return Float64(fma(Float64(-4.0 * a), c, 0.0) / Float64(fma(Float64(a * Float64(c / b)), -4.0, Float64(4.0 * b)) * a))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(-4.0 * a), $MachinePrecision] * c + 0.0), $MachinePrecision] / N[(N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] * -4.0 + N[(4.0 * b), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.6%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\left(4 \cdot a\right) \cdot c}}}{2 \cdot a} \]

   3. fp-cancel-sub-sign-inv N/A

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{b \cdot b + \left(\mathsf{neg}\left(4 \cdot a\right)\right) \cdot c}}}{2 \cdot a} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{b \cdot b} + \left(\mathsf{neg}\left(4 \cdot a\right)\right) \cdot c}}{2 \cdot a} \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, \left(\mathsf{neg}\left(4 \cdot a\right)\right) \cdot c\right)}}}{2 \cdot a} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(\mathsf{neg}\left(4 \cdot a\right)\right) \cdot c}\right)}}{2 \cdot a} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(\mathsf{neg}\left(\color{blue}{4 \cdot a}\right)\right) \cdot c\right)}}{2 \cdot a} \]

   8. distribute-lft-neg-in N/A

      \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(\left(\mathsf{neg}\left(4\right)\right) \cdot a\right)} \cdot c\right)}}{2 \cdot a} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(\left(\mathsf{neg}\left(4\right)\right) \cdot a\right)} \cdot c\right)}}{2 \cdot a} \]

   10. metadata-eval 31.7

       \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(\color{blue}{-4} \cdot a\right) \cdot c\right)}}{2 \cdot a} \]

4. Applied rewrites 31.7%

   \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)}}}{2 \cdot a} \]
5. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)}}}{2 \cdot a} \]

   2. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} + \left(-b\right)}}{2 \cdot a} \]

   3. flip-+ N/A

      \[\leadsto \frac{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} \cdot \sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} - \left(-b\right) \cdot \left(-b\right)}{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} - \left(-b\right)}}}{2 \cdot a} \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} \cdot \sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} - \left(-b\right) \cdot \left(-b\right)}{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} - \left(-b\right)}}}{2 \cdot a} \]

6. Applied rewrites 32.5%

   \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) - b \cdot b}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}}{2 \cdot a} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) - b \cdot b}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}{2 \cdot a}} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) - b \cdot b}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}}{2 \cdot a} \]

   3. associate-/l/ N/A

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) - b \cdot b}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)}} \]

   4. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) - b \cdot b}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)}} \]

   5. lift--.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) - b \cdot b}}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)} \]

   6. lift-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(\left(-4 \cdot a\right) \cdot c + b \cdot b\right)} - b \cdot b}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)} \]

   7. associate--l+ N/A

      \[\leadsto \frac{\color{blue}{\left(-4 \cdot a\right) \cdot c + \left(b \cdot b - b \cdot b\right)}}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b - b \cdot b\right)}}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)} \]

   9. +-inverses N/A

      \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot a, c, \color{blue}{0}\right)}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)} \]

   10. lower-*.f64 99.4

       \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}{\color{blue}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)}} \]

8. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)}} \]

9. Taylor expanded in a around 0

   \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}{\color{blue}{a \cdot \left(-4 \cdot \frac{a \cdot c}{b} + 4 \cdot b\right)}} \]
10. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}{\color{blue}{\left(-4 \cdot \frac{a \cdot c}{b} + 4 \cdot b\right) \cdot a}} \]

    2. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}{\color{blue}{\left(-4 \cdot \frac{a \cdot c}{b} + 4 \cdot b\right) \cdot a}} \]

    3. *-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}{\left(\color{blue}{\frac{a \cdot c}{b} \cdot -4} + 4 \cdot b\right) \cdot a} \]

    4. lower-fma.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}{\color{blue}{\mathsf{fma}\left(\frac{a \cdot c}{b}, -4, 4 \cdot b\right)} \cdot a} \]

    5. associate-/l* N/A

       \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}{\mathsf{fma}\left(\color{blue}{a \cdot \frac{c}{b}}, -4, 4 \cdot b\right) \cdot a} \]

    6. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}{\mathsf{fma}\left(\color{blue}{a \cdot \frac{c}{b}}, -4, 4 \cdot b\right) \cdot a} \]

    7. lower-/.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}{\mathsf{fma}\left(a \cdot \color{blue}{\frac{c}{b}}, -4, 4 \cdot b\right) \cdot a} \]

    8. lower-*.f64 90.9

       \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}{\mathsf{fma}\left(a \cdot \frac{c}{b}, -4, \color{blue}{4 \cdot b}\right) \cdot a} \]

11. Applied rewrites 90.9%

    \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}{\color{blue}{\mathsf{fma}\left(a \cdot \frac{c}{b}, -4, 4 \cdot b\right) \cdot a}} \]
12. Add Preprocessing

Alternative 6: 90.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, -1, -c\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (fma (/ (* (* c c) a) (* b b)) -1.0 (- c)) b))

C

double code(double a, double b, double c) {
	return fma((((c * c) * a) / (b * b)), -1.0, -c) / b;
}

Julia

function code(a, b, c)
	return Float64(fma(Float64(Float64(Float64(c * c) * a) / Float64(b * b)), -1.0, Float64(-c)) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(N[(c * c), $MachinePrecision] * a), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] * -1.0 + (-c)), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 31.6%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\left(4 \cdot a\right) \cdot c}}}{2 \cdot a} \]

   3. fp-cancel-sub-sign-inv N/A

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{b \cdot b + \left(\mathsf{neg}\left(4 \cdot a\right)\right) \cdot c}}}{2 \cdot a} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{b \cdot b} + \left(\mathsf{neg}\left(4 \cdot a\right)\right) \cdot c}}{2 \cdot a} \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, \left(\mathsf{neg}\left(4 \cdot a\right)\right) \cdot c\right)}}}{2 \cdot a} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(\mathsf{neg}\left(4 \cdot a\right)\right) \cdot c}\right)}}{2 \cdot a} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(\mathsf{neg}\left(\color{blue}{4 \cdot a}\right)\right) \cdot c\right)}}{2 \cdot a} \]

   8. distribute-lft-neg-in N/A

      \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(\left(\mathsf{neg}\left(4\right)\right) \cdot a\right)} \cdot c\right)}}{2 \cdot a} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(\left(\mathsf{neg}\left(4\right)\right) \cdot a\right)} \cdot c\right)}}{2 \cdot a} \]

   10. metadata-eval 31.7

       \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(\color{blue}{-4} \cdot a\right) \cdot c\right)}}{2 \cdot a} \]

4. Applied rewrites 31.7%

   \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)}}}{2 \cdot a} \]
5. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)}}}{2 \cdot a} \]

   2. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} + \left(-b\right)}}{2 \cdot a} \]

   3. flip-+ N/A

      \[\leadsto \frac{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} \cdot \sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} - \left(-b\right) \cdot \left(-b\right)}{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} - \left(-b\right)}}}{2 \cdot a} \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} \cdot \sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} - \left(-b\right) \cdot \left(-b\right)}{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} - \left(-b\right)}}}{2 \cdot a} \]

6. Applied rewrites 32.5%

   \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) - b \cdot b}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}}{2 \cdot a} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) - b \cdot b}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}{2 \cdot a}} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) - b \cdot b}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)}}}{2 \cdot a} \]

   3. associate-/l/ N/A

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) - b \cdot b}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)}} \]

   4. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) - b \cdot b}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)}} \]

   5. lift--.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) - b \cdot b}}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)} \]

   6. lift-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(\left(-4 \cdot a\right) \cdot c + b \cdot b\right)} - b \cdot b}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)} \]

   7. associate--l+ N/A

      \[\leadsto \frac{\color{blue}{\left(-4 \cdot a\right) \cdot c + \left(b \cdot b - b \cdot b\right)}}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b - b \cdot b\right)}}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)} \]

   9. +-inverses N/A

      \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot a, c, \color{blue}{0}\right)}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)} \]

   10. lower-*.f64 99.4

       \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}{\color{blue}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)}} \]

8. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right)\right) \cdot \left(2 \cdot a\right)}} \]

9. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{-1 \cdot c + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
10. Step-by-step derivation

    1. mul-1-neg N/A

       \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(c\right)\right)} + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b} \]

    2. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(c\right)\right) + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]

    3. +-commutative N/A

       \[\leadsto \frac{\color{blue}{-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \left(\mathsf{neg}\left(c\right)\right)}}{b} \]

    4. *-commutative N/A

       \[\leadsto \frac{\color{blue}{\frac{a \cdot {c}^{2}}{{b}^{2}} \cdot -1} + \left(\mathsf{neg}\left(c\right)\right)}{b} \]

    5. lower-fma.f64 N/A

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{a \cdot {c}^{2}}{{b}^{2}}, -1, \mathsf{neg}\left(c\right)\right)}}{b} \]

    6. lower-/.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{a \cdot {c}^{2}}{{b}^{2}}}, -1, \mathsf{neg}\left(c\right)\right)}{b} \]

    7. *-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\color{blue}{{c}^{2} \cdot a}}{{b}^{2}}, -1, \mathsf{neg}\left(c\right)\right)}{b} \]

    8. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\color{blue}{{c}^{2} \cdot a}}{{b}^{2}}, -1, \mathsf{neg}\left(c\right)\right)}{b} \]

    9. unpow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\color{blue}{\left(c \cdot c\right)} \cdot a}{{b}^{2}}, -1, \mathsf{neg}\left(c\right)\right)}{b} \]

    10. lower-*.f64 N/A

        \[\leadsto \frac{\mathsf{fma}\left(\frac{\color{blue}{\left(c \cdot c\right)} \cdot a}{{b}^{2}}, -1, \mathsf{neg}\left(c\right)\right)}{b} \]

    11. unpow2 N/A

        \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{\color{blue}{b \cdot b}}, -1, \mathsf{neg}\left(c\right)\right)}{b} \]

    12. lower-*.f64 N/A

        \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{\color{blue}{b \cdot b}}, -1, \mathsf{neg}\left(c\right)\right)}{b} \]

    13. lower-neg.f64 90.9

        \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, -1, \color{blue}{-c}\right)}{b} \]

11. Applied rewrites 90.9%

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, -1, -c\right)}{b}} \]
12. Add Preprocessing

Alternative 7: 90.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(-a, \frac{c}{b \cdot b}, -1\right)}{b} \cdot c \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (* (/ (fma (- a) (/ c (* b b)) -1.0) b) c))

C

double code(double a, double b, double c) {
	return (fma(-a, (c / (b * b)), -1.0) / b) * c;
}

Julia

function code(a, b, c)
	return Float64(Float64(fma(Float64(-a), Float64(c / Float64(b * b)), -1.0) / b) * c)
end

Wolfram

code[a_, b_, c_] := N[(N[(N[((-a) * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / b), $MachinePrecision] * c), $MachinePrecision]

Derivation

1. Initial program 31.6%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Add Preprocessing

3. Taylor expanded in c around 0

   \[\leadsto \color{blue}{c \cdot \left(-1 \cdot \frac{a \cdot c}{{b}^{3}} - \frac{1}{b}\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{a \cdot c}{{b}^{3}} - \frac{1}{b}\right) \cdot c} \]

   2. associate-*r/ N/A

      \[\leadsto \left(\color{blue}{\frac{-1 \cdot \left(a \cdot c\right)}{{b}^{3}}} - \frac{1}{b}\right) \cdot c \]

   3. associate-*r* N/A

      \[\leadsto \left(\frac{\color{blue}{\left(-1 \cdot a\right) \cdot c}}{{b}^{3}} - \frac{1}{b}\right) \cdot c \]

   4. mul-1-neg N/A

      \[\leadsto \left(\frac{\color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot c}{{b}^{3}} - \frac{1}{b}\right) \cdot c \]

   5. associate-*l/ N/A

      \[\leadsto \left(\color{blue}{\frac{\mathsf{neg}\left(a\right)}{{b}^{3}} \cdot c} - \frac{1}{b}\right) \cdot c \]

   6. distribute-neg-frac N/A

      \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\frac{a}{{b}^{3}}\right)\right)} \cdot c - \frac{1}{b}\right) \cdot c \]

   7. mul-1-neg N/A

      \[\leadsto \left(\color{blue}{\left(-1 \cdot \frac{a}{{b}^{3}}\right)} \cdot c - \frac{1}{b}\right) \cdot c \]

   8. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(-1 \cdot \frac{a}{{b}^{3}}\right) \cdot c - \frac{1}{b}\right) \cdot c} \]

5. Applied rewrites 90.6%

   \[\leadsto \color{blue}{\left(\left(-c\right) \cdot \frac{a}{{b}^{3}} - \frac{1}{b}\right) \cdot c} \]

6. Taylor expanded in b around -inf

   \[\leadsto \left(-1 \cdot \frac{1 + \frac{a \cdot c}{{b}^{2}}}{b}\right) \cdot c \]
7. Step-by-step derivation

8. Applied rewrites 90.6%

   \[\leadsto \frac{\mathsf{fma}\left(-a, \frac{c}{b \cdot b}, -1\right)}{b} \cdot c \]
9. Add Preprocessing

Alternative 8: 81.3% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \frac{-c}{b} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ (- c) b))

C

double code(double a, double b, double c) {
	return -c / b;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = -c / b
end function

Java

public static double code(double a, double b, double c) {
	return -c / b;
}

Python

def code(a, b, c):
	return -c / b

Julia

function code(a, b, c)
	return Float64(Float64(-c) / b)
end

MATLAB

function tmp = code(a, b, c)
	tmp = -c / b;
end

Wolfram

code[a_, b_, c_] := N[((-c) / b), $MachinePrecision]

Derivation

1. Initial program 31.6%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Add Preprocessing

3. Taylor expanded in a around 0

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{-1 \cdot c}{b}} \]

   2. mul-1-neg N/A

      \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(c\right)}}{b} \]

   3. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(c\right)}{b}} \]

   4. lower-neg.f64 81.2

      \[\leadsto \frac{\color{blue}{-c}}{b} \]

5. Applied rewrites 81.2%

   \[\leadsto \color{blue}{\frac{-c}{b}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024326 
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))